Relativistic X at Risk

PortfolioOptimisers.RelativisticValueatRisk Type

struct RelativisticValueatRisk{__T_settings, __T_slv, __T_alpha, __T_kappa, __T_w} <: RiskMeasure

Represents the Relativistic Value-at-Risk (RVaR) risk measure.

RelativisticValueatRisk is a coherent risk measure generalising EVaR via the Tsallis ($\kappa$-deformed) entropy. It is parametrised by a deformation parameter $\kappa \in (0,1)$ and reduces to EVaR in the limit $\kappa \to 0$. It is solved via a conic programme.

Mathematical definition

Define the $\kappa$-logarithm ${\ell }_{\kappa }(u)=\frac{u^{\kappa }-u^{-\kappa }}{2\kappa }$. The RVaR is:

$$\begin{array}{rl}{\mathrm{RVaR}}_{\alpha ,\kappa }(\mathit{x})& =\underset{t,z}{min}\{t+{\ell }_{\kappa }(\alpha T)z+\sum _{i=1}^T({\psi }_i+{\theta }_i):z\ge 0\}.\end{array}$$

Where:

• ${\mathrm{RVaR}}_{\alpha ,\kappa }(\mathit{x})$: Relativistic Value-at-Risk.

• $\mathit{x}$: Portfolio returns vector $T\times 1$.

• $\alpha$: Significance level (left tail probability), $\alpha \in (0,1)$.

• $T$: Number of observations.

• $\kappa \in (0,1)$: Tsallis deformation parameter.

• ${\ell }_{\kappa }(u)=\frac{u^{\kappa }-u^{-\kappa }}{2\kappa }$: $\kappa$-logarithm.

• $t$, $z$, ${\psi }_i$, ${\theta }_i$, ${ϵ}_i$, ${\omega }_i$: Conic optimisation variables.

subject to the power-cone constraints:

$$\begin{array}{rl}& (\frac{z(1+\kappa )}{2\kappa },\frac{{\psi }_i(1+\kappa )}{\kappa },{ϵ}_i)\in {\mathcal{K}}_{\mathrm{pow}}\left(\frac{1}{1+\kappa }\right)\mathrm{\forall }i,\\ & (\frac{{\omega }_i}{1-\kappa },\frac{{\theta }_i}{\kappa },-\frac{z}{2\kappa })\in {\mathcal{K}}_{\mathrm{pow}}(1-\kappa )\mathrm{\forall }i,\\ & {ϵ}_i+{\omega }_i\le x_i+t\mathrm{\forall }i.\end{array}$$

Where:

• ${\mathcal{K}}_{\mathrm{pow}}(p)=\{(a,b,c):a^p{b}^{1-p}\ge |c|,a\ge 0,b\ge 0\}$: Power cone.

Fields

• settings: Risk measure settings.

• slv: Solver or vector of solvers.

• alpha: Quantile level for the lower tail.

• kappa: Relativistic deformation parameter.

• w: Optional observation weights vector observations × 1, or a concrete subtype of DynamicAbstractWeights. If nothing, the computation is unweighted.

Constructors

RelativisticValueatRisk(;
    settings::RiskMeasureSettings = RiskMeasureSettings(),
    slv::Option{<:Slv_VecSlv} = nothing,
    alpha::Number = 0.05,
    kappa::Number = 0.3,
    w::Option{<:ObsWeights} = nothing
) -> RelativisticValueatRisk

Keywords correspond to the struct's fields.

Validation

• 0 < alpha < 1.

• 0 < kappa < 1.

• If slv is a VecSlv: !isempty(slv).

• If w is not nothing: !isempty(w).

Functor

(r::RelativisticValueatRisk)(x::VecNum)

Computes the RVaR of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

julia> RelativisticValueatRisk()
RelativisticValueatRisk
  settings ┼ RiskMeasureSettings
           │   scale ┼ Float64: 1.0
           │      ub ┼ nothing
           │     rke ┴ Bool: true
       slv ┼ nothing
     alpha ┼ Float64: 0.05
     kappa ┼ Float64: 0.3
         w ┴ nothing

Related

• RiskMeasure

• RiskMeasureSettings

• EntropicValueatRisk

• RelativisticValueatRiskRange

• RelativisticDrawdownatRisk

source

PortfolioOptimisers.factory Function

factory(
    r::RelativisticValueatRisk,
    pr::AbstractPriorResult;
    ...
) -> RelativisticValueatRisk
factory(
    r::RelativisticValueatRisk,
    pr::AbstractPriorResult,
    slv::Union{Nothing, Solver, AbstractVector{<:Solver}},
    args...;
    kwargs...
) -> RelativisticValueatRisk

Create an instance of RelativisticValueatRisk by selecting observation weights and solver from the risk-measure instance or falling back to the prior result.

Related

• RelativisticValueatRisk

• AbstractPriorResult

• factory

• nothing_scalar_array_selector

• solver_selector

source

PortfolioOptimisers.factory Function

factory(
    r::RelativisticValueatRisk,
    slv::Union{Solver, AbstractVector{<:Solver}};
    ...
) -> Union{RelativisticValueatRisk{RiskMeasureSettings{__T_scale, __T_ub, __T_rke}, <:AbstractVector{var"#s869"}, <:Number, <:Number} where {__T_scale, __T_ub, __T_rke, var"#s869"<:Solver}, RelativisticValueatRisk{RiskMeasureSettings{__T_scale, __T_ub, __T_rke}, Solver{__T_name, __T_solver, __T_settings, __T_check_sol, __T_add_bridges}, <:Number, <:Number} where {__T_scale, __T_ub, __T_rke, __T_name, __T_solver, __T_settings, __T_check_sol, __T_add_bridges}}
factory(
    r::RelativisticValueatRisk,
    slv::Union{Solver, AbstractVector{<:Solver}},
    pr::Union{Nothing, AbstractPriorResult},
    args...;
    kwargs...
) -> RelativisticValueatRisk

Create an instance of RelativisticValueatRisk by overriding the solver and optionally selecting observation weights from the prior result.

Related

• RelativisticValueatRisk

• AbstractPriorResult

• factory

• nothing_scalar_array_selector

• solver_selector

source

PortfolioOptimisers.RelativisticValueatRiskRange Type

struct RelativisticValueatRiskRange{__T_settings, __T_slv, __T_alpha, __T_kappa_a, __T_beta, __T_kappa_b, __T_w} <: RiskMeasure

Represents the Relativistic Value-at-Risk Range (RVaR Range) risk measure.

RelativisticValueatRiskRange computes the sum of the lower-tail RVaR (at level alpha with deformation kappa_a) and the upper-tail RVaR (at level beta with deformation kappa_b).

Mathematical definition

$$\begin{array}{rl}{\mathrm{RVaRRange}}_{\alpha ,{\kappa }_a,\beta ,{\kappa }_b}(\mathit{x})& ={\mathrm{RVaR}}_{\alpha ,{\kappa }_a}(\mathit{x})+{\mathrm{RVaR}}_{\beta ,{\kappa }_b}(-\mathit{x}).\end{array}$$

Where:

• ${\mathrm{RVaRRange}}_{\alpha ,{\kappa }_a,\beta ,{\kappa }_b}(\mathit{x})$: Relativistic VaR range.

• $\mathit{x}$: Portfolio returns vector $T\times 1$.

• ${\mathrm{RVaR}}_{\alpha ,{\kappa }_a}(\mathit{x})$: Lower-tail RVaR with parameters $(\alpha ,{\kappa }_a)$.

• ${\mathrm{RVaR}}_{\beta ,{\kappa }_b}(-\mathit{x})$: Upper-tail RVaR with parameters $(\beta ,{\kappa }_b)$.

Fields

• settings: Risk measure settings.

• slv: Solver or vector of solvers.

• alpha: Quantile level for the lower tail.

• kappa_a: Relativistic deformation parameter for the lower tail.

• beta: Quantile level for the upper tail.

• kappa_b: Relativistic deformation parameter for the upper tail.

• w: Optional observation weights vector observations × 1, or a concrete subtype of DynamicAbstractWeights. If nothing, the computation is unweighted.

Constructors

RelativisticValueatRiskRange(;
    settings::RiskMeasureSettings = RiskMeasureSettings(),
    slv::Option{<:Slv_VecSlv} = nothing,
    alpha::Number = 0.05,
    kappa_a::Number = 0.3,
    beta::Number = 0.05,
    kappa_b::Number = 0.3,
    w::Option{<:ObsWeights} = nothing
) -> RelativisticValueatRiskRange

Keywords correspond to the struct's fields.

Validation

• 0 < alpha < 1, 0 < kappa_a < 1.

• 0 < beta < 1, 0 < kappa_b < 1.

• If slv is a VecSlv: !isempty(slv).

• If w is not nothing: !isempty(w).

Functor

(r::RelativisticValueatRiskRange)(x::VecNum)

Computes the RVaR Range of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

julia> RelativisticValueatRiskRange()
RelativisticValueatRiskRange
  settings ┼ RiskMeasureSettings
           │   scale ┼ Float64: 1.0
           │      ub ┼ nothing
           │     rke ┴ Bool: true
       slv ┼ nothing
     alpha ┼ Float64: 0.05
   kappa_a ┼ Float64: 0.3
      beta ┼ Float64: 0.05
   kappa_b ┼ Float64: 0.3
         w ┴ nothing

Related

• RiskMeasure

• RiskMeasureSettings

• RelativisticValueatRisk

• EntropicValueatRiskRange

source

PortfolioOptimisers.factory Method

factory(
    r::RelativisticValueatRiskRange,
    pr::AbstractPriorResult,
    slv::Union{Nothing, Solver, AbstractVector{<:Solver}},
    args...;
    kwargs...
) -> RelativisticValueatRiskRange

Create an instance of RelativisticValueatRiskRange by selecting observation weights and solver from the risk-measure instance or falling back to the prior result.

Related

• RelativisticValueatRiskRange

• AbstractPriorResult

• factory

• nothing_scalar_array_selector

• solver_selector

source

PortfolioOptimisers.RelativisticDrawdownatRisk Type

struct RelativisticDrawdownatRisk{__T_settings, __T_slv, __T_alpha, __T_kappa, __T_w} <: RiskMeasure

Represents the Relativistic Drawdown-at-Risk (RDDaR) risk measure.

RelativisticDrawdownatRisk applies the Relativistic Value-at-Risk framework to the absolute drawdown series of portfolio returns.

Mathematical definition

Define the absolute drawdown series:

$$\begin{array}{rl}{c}_t& =\sum _{s=1}^t{x}_s,\\ d_t& =c_t-\underset{0\le s\le t}{max}{c}_s\le 0.\end{array}$$

Where:

• $\mathit{x}$: Portfolio returns vector $T\times 1$.

• $c_t$: Cumulative simple portfolio return at period $t$.

• $d_t\le 0$: Absolute drawdown at period $t$.

The Relativistic Drawdown-at-Risk is the RVaR of the drawdown series:

$$\begin{array}{rl}{\mathrm{RDDaR}}_{\alpha ,\kappa }(\mathit{x})& ={\mathrm{RVaR}}_{\alpha ,\kappa }(\mathit{d}(\mathit{x})).\end{array}$$

Where:

• ${\mathrm{RDDaR}}_{\alpha ,\kappa }(\mathit{x})$: Relativistic Drawdown-at-Risk.

• $\alpha$: Significance level (left tail probability), $\alpha \in (0,1)$.

• $\kappa \in (0,1)$: Tsallis deformation parameter.

• $\mathit{d}(\mathit{x})$: Absolute drawdown series vector $T\times 1$.

Fields

• settings: Risk measure settings.

• slv: Solver or vector of solvers.

• alpha: Quantile level for the lower tail.

• kappa: Relativistic deformation parameter.

• w: Optional observation weights vector observations × 1, or a concrete subtype of DynamicAbstractWeights. If nothing, the computation is unweighted.

Constructors

RelativisticDrawdownatRisk(;
    settings::RiskMeasureSettings = RiskMeasureSettings(),
    slv::Option{<:Slv_VecSlv} = nothing,
    alpha::Number = 0.05,
    kappa::Number = 0.3,
    w::Option{<:ObsWeights} = nothing
) -> RelativisticDrawdownatRisk

Keywords correspond to the struct's fields.

Validation

• 0 < alpha < 1.

• 0 < kappa < 1.

• If slv is a VecSlv: !isempty(slv).

• If w is not nothing: !isempty(w).

Functor

(r::RelativisticDrawdownatRisk)(x::VecNum)

Computes the Relativistic Drawdown-at-Risk of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

julia> RelativisticDrawdownatRisk()
RelativisticDrawdownatRisk
  settings ┼ RiskMeasureSettings
           │   scale ┼ Float64: 1.0
           │      ub ┼ nothing
           │     rke ┴ Bool: true
       slv ┼ nothing
     alpha ┼ Float64: 0.05
     kappa ┼ Float64: 0.3
         w ┴ nothing

Related

• RiskMeasure

• RiskMeasureSettings

• RelativisticValueatRisk

• EntropicDrawdownatRisk

• RelativeRelativisticDrawdownatRisk

source

PortfolioOptimisers.factory Function

factory(
    r::RelativisticDrawdownatRisk,
    pr::AbstractPriorResult;
    ...
) -> RelativisticDrawdownatRisk
factory(
    r::RelativisticDrawdownatRisk,
    pr::AbstractPriorResult,
    slv::Union{Nothing, Solver, AbstractVector{<:Solver}},
    args...;
    kwargs...
) -> RelativisticDrawdownatRisk

Create an instance of RelativisticDrawdownatRisk by selecting observation weights and solver from the risk-measure instance or falling back to the prior result.

Related

• RelativisticDrawdownatRisk

• AbstractPriorResult

• factory

• nothing_scalar_array_selector

• solver_selector

source

PortfolioOptimisers.factory Function

factory(
    r::RelativisticDrawdownatRisk,
    slv::Union{Solver, AbstractVector{<:Solver}};
    ...
) -> Union{RelativisticDrawdownatRisk{_A, <:AbstractVector{var"#s869"}, <:Number, <:Number} where {_A, var"#s869"<:Solver}, RelativisticDrawdownatRisk{_A, Solver{__T_name, __T_solver, __T_settings, __T_check_sol, __T_add_bridges}, <:Number, <:Number} where {_A, __T_name, __T_solver, __T_settings, __T_check_sol, __T_add_bridges}}
factory(
    r::RelativisticDrawdownatRisk,
    slv::Union{Solver, AbstractVector{<:Solver}},
    pr::Union{Nothing, AbstractPriorResult},
    args...;
    kwargs...
) -> RelativisticDrawdownatRisk

Create an instance of RelativisticDrawdownatRisk by overriding the solver and optionally selecting observation weights from the prior result.

Related

• RelativisticDrawdownatRisk

• AbstractPriorResult

• factory

• nothing_scalar_array_selector

• solver_selector

source

PortfolioOptimisers.RelativeRelativisticDrawdownatRisk Type

struct RelativeRelativisticDrawdownatRisk{__T_settings, __T_slv, __T_alpha, __T_kappa, __T_w} <: HierarchicalRiskMeasure

Represents the Relative Relativistic Drawdown-at-Risk (Relative RDDaR) risk measure for hierarchical optimisation.

RelativeRelativisticDrawdownatRisk applies the Relativistic Value-at-Risk framework to the relative (compounded) drawdown series of portfolio returns.

Mathematical definition

Define the compounded wealth process and relative drawdown series:

$$\begin{array}{rl}{C}_t& =\prod _{s=1}^t(1+x_s),\\ r{d}_t& =\frac{C_t}{\underset{0\le s\le t}{max}{C}_s}-1\le 0.\end{array}$$

Where:

• $\mathit{x}$: Portfolio returns vector $T\times 1$.

• $C_t$: Compound wealth process at period $t$.

• $r{d}_t\le 0$: Relative drawdown at period $t$.

The Relative Relativistic Drawdown-at-Risk is the RVaR of the relative drawdown series:

$$\begin{array}{rl}{\mathrm{RRDDaR}}_{\alpha ,\kappa }(\mathit{x})& ={\mathrm{RVaR}}_{\alpha ,\kappa }(\mathit{r}\mathit{d}(\mathit{x})).\end{array}$$

Where:

• ${\mathrm{RRDDaR}}_{\alpha ,\kappa }(\mathit{x})$: Relative Relativistic Drawdown-at-Risk.

• $\alpha$: Significance level (left tail probability), $\alpha \in (0,1)$.

• $\kappa \in (0,1)$: Tsallis deformation parameter.

• $\mathit{r}\mathit{d}(\mathit{x})$: Relative drawdown series vector $T\times 1$.

Fields

• settings: Risk measure settings.

• slv: Solver or vector of solvers.

• alpha: Quantile level for the lower tail.

• kappa: Relativistic deformation parameter.

• w: Optional observation weights vector observations × 1, or a concrete subtype of DynamicAbstractWeights. If nothing, the computation is unweighted.

Constructors

RelativeRelativisticDrawdownatRisk(;
    settings::HierarchicalRiskMeasureSettings = HierarchicalRiskMeasureSettings(),
    slv::Option{<:Slv_VecSlv} = nothing,
    alpha::Number = 0.05,
    kappa::Number = 0.3,
    w::Option{<:ObsWeights} = nothing
) -> RelativeRelativisticDrawdownatRisk

Keywords correspond to the struct's fields.

Validation

• 0 < alpha < 1.

• 0 < kappa < 1.

• If slv is a VecSlv: !isempty(slv).

• If w is not nothing: !isempty(w).

Functor

(r::RelativeRelativisticDrawdownatRisk)(x::VecNum)

Computes the Relative Relativistic Drawdown-at-Risk of a portfolio returns vector x.

Arguments

• x::VecNum: Portfolio returns vector.

Examples

julia> RelativeRelativisticDrawdownatRisk()
RelativeRelativisticDrawdownatRisk
  settings ┼ HierarchicalRiskMeasureSettings
           │   scale ┴ Float64: 1.0
       slv ┼ nothing
     alpha ┼ Float64: 0.05
     kappa ┼ Float64: 0.3
         w ┴ nothing

Related

• HierarchicalRiskMeasure

• HierarchicalRiskMeasureSettings

• RelativisticDrawdownatRisk

• RelativeEntropicDrawdownatRisk

source

PortfolioOptimisers.factory Function

factory(
    r::RelativeRelativisticDrawdownatRisk,
    pr::AbstractPriorResult;
    ...
) -> RelativeRelativisticDrawdownatRisk
factory(
    r::RelativeRelativisticDrawdownatRisk,
    pr::AbstractPriorResult,
    slv::Union{Nothing, Solver, AbstractVector{<:Solver}},
    args...;
    kwargs...
) -> RelativeRelativisticDrawdownatRisk

Create an instance of RelativeRelativisticDrawdownatRisk by selecting observation weights and solver from the risk-measure instance or falling back to the prior result.

Related

• RelativeRelativisticDrawdownatRisk

• AbstractPriorResult

• factory

• nothing_scalar_array_selector

• solver_selector

source

PortfolioOptimisers.factory Function

factory(
    r::RelativeRelativisticDrawdownatRisk,
    slv::Union{Solver, AbstractVector{<:Solver}};
    ...
) -> Union{RelativeRelativisticDrawdownatRisk{HierarchicalRiskMeasureSettings{__T_scale}, <:AbstractVector{var"#s869"}, <:Number, <:Number} where {__T_scale, var"#s869"<:Solver}, RelativeRelativisticDrawdownatRisk{HierarchicalRiskMeasureSettings{__T_scale}, Solver{__T_name, __T_solver, __T_settings, __T_check_sol, __T_add_bridges}, <:Number, <:Number} where {__T_scale, __T_name, __T_solver, __T_settings, __T_check_sol, __T_add_bridges}}
factory(
    r::RelativeRelativisticDrawdownatRisk,
    slv::Union{Solver, AbstractVector{<:Solver}},
    pr::Union{Nothing, AbstractPriorResult},
    args...;
    kwargs...
) -> RelativeRelativisticDrawdownatRisk

Create an instance of RelativeRelativisticDrawdownatRisk by overriding the solver and optionally selecting observation weights from the prior result.

Related

• RelativeRelativisticDrawdownatRisk

• AbstractPriorResult

• factory

• nothing_scalar_array_selector

• solver_selector

source

PortfolioOptimisers.RRM Function

RRM(x, slv, alpha = 0.05, kappa = 0.3, ...; kwargs...)

Compute the Relativistic Risk Measure (RRM) for a vector of portfolio returns.

Solves a convex optimisation problem to compute the RRM at confidence level alpha with relativistic parameter kappa, using the specified solver(s).

Arguments

• x: Vector of portfolio returns.

• slv: Solver or vector of solvers.

• alpha: Confidence level (default 0.05).

• kappa: Relativistic parameter (default 0.3).

• Additional parameters depending on the specific RRM formulation.

• kwargs...: Additional keyword arguments passed to the solver.

Returns

• RRM value (scalar).

Related

• RelativisticValueatRisk

• Slv_VecSlv

source